Download a Gröbner Basis?

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W H A T
I S . . .
a Gröbner Basis?
Bernd Sturmfels
A Gröbner basis is a set of multivariate polynomials that has desirable algorithmic properties. Every
set of polynomials can be transformed into a Gröbner basis. This process generalizes three familiar
techniques: Gaussian elimination for solving linear
systems of equations, the Euclidean algorithm for
computing the greatest common divisor of two
univariate polynomials, and the Simplex Algorithm
for linear programming; see [3]. For example, the
input for Gaussian elimination is a collection of
linear forms such as
F = 2x + 3y + 4z − 5, 3x + 4y + 5z − 2 ,
and the algorithm transforms F into the Gröbner
basis
G = x − z + 14, y + 2z − 11 .
Let K be any field, such as the real numbers
K = R , the complex numbers K = C , the rational
numbers K = Q , or a finite field K = Fp . We write
K[x1 , . . . , xn ] for the ring of polynomials in n variables xi with coefficients in the field K . If F is any
set of polynomials, then the ideal generated by F
is the set F consisting of all polynomial linear
combinations:
F = p1 f1 + · · · + pr fr : f1 , . . . , fr ∈ F
and p1 , . . . , pr ∈ K[x1 , . . . , xn ] .
In our example the set F and its Gröbner basis G
generate the same ideal: G = F. By Hilbert’s
Basis Theorem, every ideal I in K[x1 , . . . , xn ] has the
form I = F ; i.e., it is generated by some finite set
F of polynomials.
A term order on K[x1 , . . . , xn ] is a total order ≺
a
a
on the set of all monomials xa = x11 · · · xnn which
has the following two properties:
(1) It is multiplicative; i.e., xa ≺ xb implies
a+c
x
≺ xb+c for all a, b, c ∈ Nn .
(2) The constant monomial is the smallest; i.e.,
1 ≺ xa for all a ∈ Nn \{0} .
Bernd Sturmfels is a professor of mathematics and computer science at the University of California at Berkeley.
His email address is [email protected].
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An example of a term order (for n = 2) is the
degree lexicographic order
1 ≺ x1 ≺ x2 ≺ x21 ≺ x1 x2 ≺ x22 ≺ x31 ≺ x21 x2 ≺ · · · .
If we fix a term order ≺ , then every polynomial
f has a unique initial term in≺ (f ) = xa . This is the
≺ -largest monomial xa which occurs with nonzero
coefficient in the expansion of f . We write the terms
of f in ≺ -decreasing order, and we often underline
the initial term. For instance, a quadratic polynomial is written
f = 3x22 + 5x1 x2 + 7x21 + 11x1 + 13x2 + 17.
Suppose now that I is an ideal in K[x1 , . . . , xn ] .
Then its initial ideal in≺ (I) is the ideal generated
by the initial terms of all the polynomials in I :
in≺ (I) = in≺ (f ) : f ∈ I.
A finite subset G of I is a Gröbner basis with respect to the term order ≺ if the initial terms of the
elements in G suffice to generate the initial ideal:
in≺ (I) = in≺ (g) : g ∈ G .
There is no minimality requirement for being a
Gröbner basis. If G is a Gröbner basis for I , then
any finite subset of I that contains G is also a Gröbner basis. To remedy this nonminimality, we say
that G is a reduced Gröbner basis if
(1) for each g ∈ G , the coefficient of in≺ (g) in g
is 1 ,
(2) the set {in≺ (g) : g ∈ G} minimally generates
in≺ (I) , and
(3) no trailing term of any g ∈ G lies in in≺ (I) .
With this definition, we have the following theorem:
If the term order ≺ is fixed, then every ideal I in
K[x1 , . . . , xn ] has a unique reduced Gröbner basis.
The reduced Gröbner basis G can be computed
from any generating set of I by a method that was
introduced in Bruno Buchberger’s 1965 dissertation.
Buchberger named his method after his advisor,
Wolfgang Gröbner. With hindsight, the idea of Gröbner bases can be traced back to earlier sources, including a paper written in 1900 by the invariant theorist Paul Gordan. But Buchberger was the first to
give an algorithm for computing Gröbner bases.
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VOLUME 52, NUMBER 10
Gröbner bases are very useful for solving systems of polynomial equations. Suppose K ⊆ C , and
let F be a finite set of polynomials in K[x1 , . . . , xn ] .
The variety of F is the set of all common complex
zeros:
V (F ) = (z1 , . . . , zn ) ∈ Cn : f (z1 , . . . , zn ) = 0
for all f ∈ F .
The variety does not change if we replace F by another set of polynomials that generates the same
ideal in K[x1 , . . . , xn ] . In particular, the reduced
Gröbner basis G for the ideal F specifies the
same variety:
V (F ) = V (F) = V (G) = V (G).
The advantage of G is that it reveals geometric
properties of the variety that are not visible from
F . The first question that one might ask about a
variety V (F ) is whether it is empty. Hilbert’s Nullstellensatz implies that
the variety V (F ) is empty if
and only if G equals {1}.
How can one count the number of zeros of a
given system of equations? To answer this, we
need one more definition. Given a fixed ideal I in
K[x1 , . . . , xn ] and a term order ≺ , a monomial
a
a
xa = x11 · · · xnn is called standard if it is not in the
initial ideal in≺ (I) . The number of standard monomials is finite if and only if every variable xi appears
to some power in the initial ideal. For example, if
in≺ (I) = x31 , x42 , x53 , then there are sixty standard
monomials, but if in≺ (I) = x31 , x42 , x1 x43 , then the
set of standard monomials is infinite.
The variety V (I) is finite if and only if the set
of standard monomials is finite, and the number
of standard monomials equals the cardinality of
V (I) , when zeros are counted with multiplicity.
For n = 1 this is the Fundamental Theorem of Algebra, which states that the variety V (f ) of a univariate polynomial f ∈ K[x] of degree d consists of
d complex numbers. Here the singleton {f } is a
Gröbner basis, and the standard monomials are
1, x, x2 , . . . , xd−1 .
Our criterion for deciding whether a variety is
finite generalizes to the following formula for the
dimension of a variety. Consider a subset S of the
variables {x1 , . . . , xn } such that no monomial in the
variables in S appears in in≺ (I) , and suppose that
S has maximal cardinality among all subsets with
this property. That maximal cardinality |S| equals
the dimension of V (I) .
The set of standard monomials is a K -vectorspace basis for the residue ring K[x1 , . . . , xn ]/I .
The image of a polynomial p modulo I can be
expressed uniquely as a K -linear combination of
standard monomials. This expression is the normal
form of p. The process of computing the normal
NOVEMBER 2005
form is the division algorithm. In the familar
case of only one variable x , where I = f and f
has degree d , the division algorithm writes any
polynomial p ∈ K[x] as a K -linear combination of
1, x, x2 , . . . , xd−1 . But the division algorithm works
relative to any Gröbner basis G in any number of
variables.
How can we test whether a given set of polynomials G is a Gröbner basis or not? Consider any two
polynomials g and g in G , and form their S-polynomial mg − mg . Here m and m are monomials
of smallest possible degree such that m·in≺ (g) =
m·in≺ (g ) . The S -polynomial mg − mg lies in the
ideal G . We apply the division algorithm with respect to the tentative Gröbner basis G to mg − mg .
The resulting normal form is a K -linear combination of monomials none of which is divisible by an
initial monomial from G . A necessary condition
for G to be a Gröbner basis is
normalformG (mg − mg ) = 0 for all g, g ∈ G.
Buchberger’s Criterion states that this necessary
condition is sufficient: a set G of polynomials is a
Gröbner basis if and only if all its S -polynomials
have normal form zero. From this criterion, one
derives Buchberger’s Algorithm [1] for computing
the reduced Gröbner basis G from any given input
set F .
In summary, Gröbner bases and the Buchberger
Algorithm for finding them are fundamental notions in algebra. They furnish the engine for more
advanced computations in algebraic geometry,
such as elimination theory, computing cohomology,
resolving singularities, etc. Given that polynomial
models are ubiquitous across the sciences and engineering, Gröbner bases have been used by researchers in optimization, coding, robotics, control
theory, statistics, molecular biology, and many
other fields. We invite the reader to experiment with
one of the many implementations of Buchberger’s
algorithm (e.g., in CoCoA, Macaulay2, Magma, Maple,
Mathematica, or Singular).
References
[1] DAVID COX, JOHN LITTLE, and DONAL O’ SHEA, Ideals, Varieties and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra,
second ed., Undergraduate Texts in Mathematics,
Springer-Verlag, New York, 1997.
[2] NIELS LAURITZEN, Concrete Abstract Algebra: From Numbers to Gröbner Bases, Cambridge University Press,
2003.
[3] BERND STURMFELS, Two Lectures on Gröbner Bases, New
Horizons in Undergraduate Mathematics, VMath Lecture Series, Mathematical Sciences Research Institute,
Berkeley, California, 2005, http://www.msri.org/
communications/vmath/special_productions/.
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